Data Analysis and Statistics for Geography, Environmental Science, and Engineering

29

2

Probability Theory

2.1 EVENTS AND PROBABILITIES

Probability theory is the basis for the analysis of uncertainty in science and engineering. The con-

cept of probability ties to a numerical measure of the likelihood of an event, which is one out-

come of an experiment or measurement. The sample space is the set of all possible outcomes, and

therefore an event is a subset of the sample space. Probability can thus be dened as a real number

between zero and one (0 and 1 included) assigned to the likelihood of an event. As a shorthand for

the probability of an event, we can write Pr[event] or P[event]. For example, for event A, Pr[A] or

P[A], is a real number between 0 and 1 (0 and 1 included).

It is common to give examples of games of chance when illustrating probability. Consider

rolling a six-sided die. The sample space has six possible outcomes, U = {side facing up is

1, side facing up is 2, …, side facing up is 6}. Note the use of curly brackets to dene set of

events. Dene event A = {side facing up is number 3}, then P[A] = 1/6 or 1 out of 6 possible

and equally likely outcomes.

2.2 ALGEBRA OF EVENTS

For didactic purposes, events are usually illustrated using Venn diagrams and set theory. Events are

represented by shapes or areas located in a box or domain. The universal event is the sample space

U (includes all possible events), and therefore occurs with absolute certainty P[U] = 1. For example,

U = {any number 1 to 6 faces up after rolling a die}. See Figure 2.1.

The null event is an impossible event or one that includes none of the possible events. Therefore,

its probability is zero, P[ϕ] = 0. For example, ϕ = {the side with number 0 will face up}. This is not

possible because the die does not have a side with number 0.

An oval shape represents an event A within U as shown in Figure 2.1. We also refer to B as the

complement of A, i.e., the only other event that could occur. Therefore, the only outcomes are that

A happens or B happens. Also, A and B are mutually exclusive and collectively exhaustive. The

complement is an important concept often used to simplify solving problems. It is the same as B is

NOT A, which in shorthand is B = Ā where the bar on top of the event means complement or logical

operation NOT.

In the Venn diagram of Figure 2.1, B is shaded. The box represents U and the clear oval repre-

sents A. The key numeric relation is

PB PA

[] = 1 [ ]−

(2.1)

Also, note that the complement of U is the null event.

Example from rolling a six-sided die, dene B = {any side up except a six}, A = {side sixfacesup},

determine P[B]. Solution: rst note that B = Ā and therefore we can use P[B] = 1 − 1/6 = 5/6.

Wedid not have to enumerate B with detail, just subtracted from 1.